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First question: There are plenty of "standard" examples, for instance Eisenstein series, theta series, and eta products (doing a web search for any of these will bring up a lot of information). Secon …

I remember once asking Rob Pollack, at a conference in Oxford, what the running time of his algorithm to compute overconvergent modular symbols was. His response was, "We have an algorithm?". He subse …

"Currently known" is absurd; there is an explicit formula for $\operatorname{dim} S_2(N)$ and from this it is an elementary exercise to find all such $N$. There are exactly fifteen of them, namely $\{ …

Let $\Gamma = \Gamma_0(2)$, and let $\Delta$ be the usual weight 12 cusp form. Then $f(z) = \Delta(2z) / \Delta(z)$ is a meromorphic modular function of weight 0 and level $\Gamma$, holomorphic on the …

I'm wary of speaking about "the trivialisation of $\omega$", since the unit group $\mathcal{O}(Y(N))^\times$ (the group of modular units) is a pretty big group, and your trivialization will only be we …

[Comment reposted as an answer] Up to the scaling by $2 \zeta(2s)$, which is just a matter of conventions, both are special cases of a single more general object: the series $$ E_k(z, s) = \sum_{(c, …

You don't define your notations, as Olivier points out, but I'm going to assume that $V_p$ is the map $S_k(\Gamma_0(N)) \to S_k(\Gamma_0(Np))$ given by $f(z) \mapsto f(pz)$. Then the answer is "yes" …

You can do this in Sage but "in disguise". The idea is that if $f(z)$ is a cusp form for $\Gamma(N)$, then $g(z) := f(Nz)$ is a cusp form for a certain subgroup intermediate between $\Gamma_0(N^2)$ an …

Let $N \ge 1$ and let $\ell$ and $p$ be primes not dividing $N$. The classical Ihara lemma says that if $Y_1(N, \ell)$ is the modular curve attached to the subgroup $\Gamma_1(N) \cap \Gamma_0(\ell)$, …

An explicit formula is given at the bottom of page 21 of this PhD thesis.

I'm going to add an answer to my own question, not because there's anything wrong with user94346's answer, but because I found a paper in the literature treating exactly this question: Erez Lapid and …

Let's bound the level of such an $f$ in two stages. Firstly, let's look at a prime $\ell \ne p$. Here there is a theorem of Livne and (independently) Carayol which says that if $\rho$ is a lifting of …

Try looking in the book by Bellaiche and Chenevier, "Families of Galois representations and higher rank Selmer groups" (published in Asterisque, or available from the Arxiv here). They give a nice des …

I personally would say that the root of the problem is the absence of a globally defined exponential map on $\mathbb{C}_p$. In the complex world $z \mapsto q(z) = \exp(2 \pi i z)$ is an isomorphism be …

The space $M_k(N)$ has a basis in $\mathbb{Z}[[q]]$ for any $N$ and $k$. This is a straightforward consequence of Eichler-Shimura theory and will be in any decent textbook on modular forms (e.g. Diamo …